Geometric Regularity Criteria for Incompressible Navier--Stokes Equations with Navier Boundary Conditions

TL;DR

This paper establishes geometric regularity criteria for weak solutions to 3D incompressible Navier--Stokes equations with boundary effects, focusing on vortex structure alignment and boundary conditions.

Contribution

It introduces boundary regularity theorems considering vortex geometry and boundary effects for Navier--Stokes equations with classical boundary conditions.

Findings

Boundary regularity theorem proved for regular domains.

Vorticity coherence implies regularity at boundaries.

Results apply to round balls, half-spaces, and cylindrical ducts.

Abstract

We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on regular domains with a class of oblique derivative boundary conditions, providing that the vorticity of the fluid is coherently aligned. In particular, we establish the boundary regularity on round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.